Local news outlets reported that three people, including a member of the family, had been arrested as part of a new police inquiry into influence-peddling.

Political experts have pointed out that by vacating the presidency, Mr. Zuma would make himself vulnerable to pending inquiries and corruption charges that he has been able to deflect as the sitting president.

The developments, as well as conflicting messages from leaders of the A.N.C., deepened the uncertainty surrounding Mr. Zuma’s future and the paralysis over South Africa’s governing party and government.

They further complicated efforts by Mr. Zuma’s presumed successor, Cyril Ramaphosa, the deputy president, to achieve a smooth transfer of power that would avoid widening fissures inside the party.

The recent events were also a clear sign of how much has changed in the two months since Mr. Ramaphosa was chosen to succeed Mr. Zuma as the leader of the A.N.C., creating what South Africans refer to as the two centers of power.

Mr. Zuma, seemingly untouchable just a couple of months ago, is now almost certain to lose the presidency, either through a resignation or through a vote in Parliament.

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Police officers closed off roads around the Johannesburg home of the Gupta family, close allies of President Jacob Zuma of South Africa.Credit James Oatway/Reuters

The police’s investigative unit — which has long been subject to political interference — is now investigating the Guptas, a powerful family who appeared to operate above the law under Mr. Zuma’s protection.

Analysts said it was no coincidence that heavily armed police officers had raided the Guptas’ luxury compound and carried out arrests even as Mr. Zuma appeared to dither over whether to address the nation. The intended message, they said, was that those closest to Mr. Zuma, or even Mr. Zuma himself, could be next unless he acceded to the party’s order to quit, well before his term as president was scheduled to expire in mid-2019.

Ace Magashule, the party’s secretary general, said that Mr. Zuma had been given no deadline, but the party was clearly divided over how forcefully to push the president to the exit.

Mr. Magashule, a longtime ally of the president, said that the party was not preparing a motion of no confidence against him in Parliament if he refused to resign. But other high-ranking officials issued stronger warnings, with Gwede Mantashe, the party’s chairman and an ally of Mr. Ramaphosa, said that Mr. Zuma had been given the chance to resign on his own.

“Once you resist, we are going to let you be thrown out through the vote of no confidence, because you disrespect the organization and you disobey it,” Mr. Mantashe said. “Therefore we are going to let you be devoured by the vultures.”

Around noon on Wednesday, before Mr. Zuma spoke, A.N.C. officials held a news conference announcing plans for the no-confidence motion. They said they would work with an opposition party, the Economic Freedom Fighters, which had already pushed for a no-confidence vote in Parliament.

In the interview, which came after the announcement of the no-confidence vote, Mr. Zuma said he had offered to step down — though not before June. Until then, he said, he needed to carry out certain duties, including introducing his successor, Mr. Ramaphosa, to other African heads of state at regional meetings.

He said he had told party officials: “I should agree to resign, but let us work on a time frame.” A.N.C. officials have said previously that Mr. Zuma pressed to remain in office for three to six months.

The A.N.C.’s difficult position was on clear display the previous day. At a news conference at its headquarters in Johannesburg, Mr. Magashule — who is third in the party’s hierarchy and who has traditionally acted as its spokesman — struggled to explain why the party was asking for Mr. Zuma’s resignation.

Mr. Magashule said that the ethical challenges that the president was facing had played no role, saying, “We did not take these decisions because Comrade Jacob Zuma has done anything wrong.”

Mr. Magashule’s remarks suggested the party might be reluctant to deal head-on with the culture of corruption that was endemic under Mr. Zuma.

The reason he gave for the party’s move was that Mr. Zuma’s continued presence as the nation’s leader would “erode the renewed hope and confidence among South Africans” since party elections in December, in which Mr. Ramaphosa defeated Mr. Zuma’s preferred candidate for the leadership of the A.N.C.

Mr. Magashule indicated that Mr. Zuma was hurting the party’s electoral prospects, a point that Mr. Ramaphosa’s allies had emphasized. In the 2016 local elections, the A.N.C. lost control over the nation’s biggest cities after it was deserted by traditional supporters disillusioned by Mr. Zuma’s conduct; some party officials have since warned that it might face a similar fate in national elections in 2019.